Quotations in Context: Jacobi

“Man muss immer generalisieren.” [One must always generalize.]

“Man muss immer umkehren.” [One must always invert.]

The above quotations frequently appear in modern publications, both in works of mathematics as well as in other subject areas; for example, the second quotation seems to be particularly popular in the field of finance. While all of these modern sources attribute both sentences to the nineteenth century mathematician Carl Jacobi, none of them provide a primary source or any original context for the quotations.

Photograph of Jacobi. Convergence Portrait Gallery.

The oldest work that I’ve managed to locate that contains either of these quotations is “Current Tendencies of Mathematical Research” by Edward B. Van Vleck, which appeared in the October 1916 issue of the Bulletin of the American Mathematical Society. Professor Van Vleck was a faculty member at the University of Wisconsin-Madison and a past AMS president (1913–1914), and this paper contains an address he presented during the University of Chicago’s Quarter-Centennial celebration.

Photograph of Van Vleck. Convergence Portrait Gallery.

The paper began by exploring examples of how specific problems provided influence and direction to the development of mathematics during various historical periods, which led Van Vleck to ask the question, “What are the central problems in the mathematical research of to-day?” [Van Vleck 1916, p. 2]. He offered the topic of infinite sets as one possible answer to the question. It was at this point in the paper that the second quotation of this column appears:

The characteristic tendency in the thought of to-day which I have tried to grasp under the comprehensive term “Problem of the infinite set” is shown rather as a current beneath the surface than in any individual concrete problem. The average investigator must perforce seize upon any problem which his brains find at hand. There exist, however, certain fundamental principles which will aid him in finding a worthy one. The great mathematician Jacobi is said to have inculcated upon his students the dictum: Man muss immer umkehren. One must always seek a converse, turn a thought the other end to. It was by turning the elliptic integral inside out that Jacobi obtained his splendid theory of elliptic and theta functions [Van Vleck 1916, p. 3].

In the very next paragraph of the paper, the first quotation of this column appeared; however, in this case Van Vleck did not attribute it to Jacobi, but instead claimed it as his own invention:

Without dwelling further upon the fertility of Jacobi’s dictum, I wish to coin and put beside it another obvious dictum of yet wider reach: Man muss immer generalizieren [Van Vleck 1916, p. 3].

Later in the paper, Van Vleck repeated both quotations and again distinctly gave Jacobi credit for “Man muss immer umkehren,” but not for “Man muss immer generalizieren.” While it is possible that Jacobi used both phrases without Van Vleck being aware of it, the contents of this paper and the lack of any apparent earlier sources both suggest that Van Vleck is the true, original source of the “generalize” version of the quotation.

As for “Man muss immer umkehren,” while Van Vleck did attribute it to Jacobi, he provided no details on his source for the quotation; further, since he only claimed that Jacobi “is said to have” made such a statement, it seems clear that Van Vleck didn’t derive the quote directly from any of Jacobi’s works. While I haven’t been able to examine every single book, paper, or letter written by Jacobi, my inability to find any source prior to 1916 that contains even a passing reference to this quotation raises the possibility that it also may be misattributed and not truly originate with Jacobi.

Reference

Van Vleck, Edward B. 1916, October. “Current Tendencies of Mathematical Research.” Bulletin of the American Mathematical Society 23(1): 1–13.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the CSHPM/SCHPM Bulletin of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in November 2016). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.